Archive for November, 2016

We’ve been discussing the mechanical advantage that compound pulleys provide to humans during lifting operations and last time we hit upon the fact that there comes a point of diminished return, a reality that engineers must negotiate in their mechanical designs. Today we’ll discuss one of the undesirable tradeoffs that results in a diminished return within a compound pulleyarrangement when we compute the length of rope the Grecian man we’ve been following must grapple in order to lift his urn. What we’ll discover is a situation of mechanical overkill – like using a steamroller to squash a bug.

As presented in a past blog, the equations for work input, WI, and work output, WO, we’ll be using are,

WI = F × d2

WO = W × d1

Now, ideally, in a compound pulley no friction exists in the wheels to impede the rope’s movement, and that will be our scenario today. Our next blog will deal with the more complex situation where friction is present. So for our example today, with no friction present, work input equals output…

WI = WO

… and this fact allows us to develop an equation in terms of the rope length/distance factors in our compound pulley assembly, represented by d1 and d2, …

F × d2 = W × d1

d2 ÷ d1 = W ÷ F

Now, from our last blog we know that W divided by F represents the mechanical advantage, MA, to Mr. Toga of using the compound pulley, which was found to be 16, equivalent to the sections of rope directly supporting the urn. We’ll set the distance factors up in relation to MA, and the equation becomes…

d2÷ d1 = MA

d2 = MA × d1

d2 = 16 ×2 feet = 32 feet

What we discover is that in order to raise the urn 2 feet, our Grecian friend must manipulate 32 feet of rope – which would only make sense if he were lifting something far heavier than a 40 pound urn.

In reality, WI does not equal WO, due to the inevitable presence of friction. Next time we’ll see how friction affects the mechanical advantage in our compound pulley.

We’re all familiar with the phrase, “too much of a good thing.” As a professional engineer, I’ve often found this to be true. No matter the subject involved, there inevitably comes a point when undesirable tradeoffs occur. We’ll begin our look at this phenomenon in relation to compound pulleys today, and we’ll see how the pulley arrangement we’ve been working with encounters a rope length tradeoff. Today’s arrangement has a lot of pulleys lifting an urn a short distance.

We’ll be working with two distance/length factors and observe what happens when the number of pulleys is increased. Last time we saw how the compound pulley is essentially a work input-output device, which makes use of distance factors. In our example below, the first distance/length factor, d1, pertains to the distance the urn is lifted above the ground. The second factor, d2, pertains to the length of rope Mr. Toga extracts from the pulley while actively lifting. It’s obvious that some tradeoff has occurred just by looking at the two lengths of rope in the image below as compared to last week. What we’ll see down the road is that this also affects mechanical advantage.

The compound pulley here consists of 16 pulleys, therefore it provides a mechanical advantage, MA, of 16. For a refresher on how MA is determined, see our preceding blog.

Rope Length Tradeoff in a Compound Pulley

With an MA of 16 and the urn’s weight, W, at 40 pounds, we compute the force, F, Mr. Toga must exert to actively lift the urn higher must be greater than,

F >W ÷ MA

F > 40 Lbs. ÷16

F > 2.5 Lbs.

Although the force required to lift the urn is a small fraction of the urn’s weight, Mr. Toga must work with a long and unwieldy length of rope. How long? We’ll find out next time when we’ll take a closer look at the relationship between d1 and d2.

In our last blog we saw how adding extra pulleys resulted in mechanical advantagebeing doubled, which translates to a 50% decreased lifting effort over a previous scenario.Pulleys are engineering marvels that make our lives easier. Theoretically, the more pulleys you add to a compound pulley arrangement, the greater the mechanical advantage — up to a point. Eventually you’d encounter undesirable tradeoffs. We’ll examine those tradeoffs, but before we do we’ll need to revisit the engineering principle of work and see how it applies to compound pulleys as a work input-output device.

Pulleys as a Work Input-Outut Device

The compound pulley arrangement shown includes distance notations, d1 and d2. Their inclusion allows us to see it as a workinput-output device.Work is input by Mr. Toga, we’ll call that WI, when he pulls his end of the rope using his bicep force, F. In response to his efforts, workis output by the compound pulley when the urn’s weight, W, is lifted off the ground against the pull of gravity. We’ll call that work outputWO.

In a previous blog we defined work as a factor of force multiplied by distance. Using that notation, when Mr. Toga exerts a force F to pull the rope a distance d2 , his work input is expressed as,

WI = F × d2

When the compound pulley lifts the urn a distance d1 above the ground against gravity, its work output is expressed as,

WO = W × d1

Next time we’ll compare our pulley’swork input to output to develop a relationship between d1 and d2. This relationship will illustrate the first undesirable tradeoff of adding too many pulleys.